Month: June 2012

This task isn’t worth much if you don’t start with intuition. You should point to this image and ask your students to intuit the location of a fair horizontal cut. At the moment, I think my best option is to print out that frame and pass it out to students so they can each draw their own lines. What I need, though, is a digital system where students can adjust that line precisely to their liking and then tap submit.

After that, the students see a composite of their classmates’ guesses.

This does two things. One, it ratchets up engagement. We want to know what the answer is and who guessed closest. Two, the mathematical model gets a lot of credibility when its solution falls right in the middle of our field of guesses.

This task isn’t worth much if you don’t end with generalization. The initial task sets the hook, but it resolves quickly into computation. Where this task (and others like it) light up the board is when we say, “Okay, now tell me where to make the cut for any size wedge of cheese. Any angle. Any radius.”

The ideal outcome on a digital device is that the student comes up with an abstract function with respect to theta and r, enters it into the device, and then that abstraction gets concretized right on the original image. The student sees the result of her model on a dynamic cheese wedge. She adjusts the theta slider and sees both the wedge and the cut adjust dynamically according to her function.

That’s the ladder of abstraction right there — from intuition to generalization.

Featured Commentary

There’s an interesting back-and-forth in the comments with one side claiming that the obviousness of the vertical cut makes the horizontal cut kind of contrived and another side saying it doesn’t matter.

There’s always the critique that Khan Academy is not pedagogically sound, that we’re procedural-based, focusing on mechanics without base understanding but I actually think we’re the exact opposite of that.

[..]

With procedural, worked problems: That’s how I learned, that’s how everyone I knew learned. But we do have videos explaining the ‘why’ of things, like borrowing, or highly rigorous concepts like college-level linear algebra, so it’s kind of weird when people are nitpicking about multiplying negative numbers.

Maybe something got lost in the edit, but I can’t seem to reconcile those two statements. On one line, Khan Academy is the opposite of procedural learning. In the next paragraph, Khan offers a full-throated endorsement of procedural learning through worked examples.

We will never say that our visual library is perfect. And we’re constantly trying to improve. But I think it’s a straw-man argument to pick one video and say, ‘This is a procedural video, it is not conceptual, they’re all like this, these people don’t have an understanding of pedagogy.’ That is, frankly, a bit arrogant and disparaging.

The statement “this should have been better” isn’t the same as “this should have been perfect.” Khan has god-knows-how-many videos at this point, some of which he made with only his cousins in mind, and we should expect a wide distribution of quality.

Setting aside any of our concerns about the best place for video lectures in a math classroom, we all have an interest in Khan’s video lectures being as mathematically correct as possible. But Khan thinks it’s arrogant and disparaging for people who have spent decades witnessing and cataloging every possible misconception about negative numbers to step in and say, “Your video may lead to misconceptions about negative numbers.” That’s a pity. I encourage Khan and his staff to find a more productive way to engage this deep bench of unpaid, well-informed critics.

BTW. If Khan is wondering why math teachers worry about his pedagogical content knowledge, this is the sort of decision that gives us the heebie-jeebies:

Mr. Khan says he intentionally mixed up the transitive and associative properties to show that understanding that a times b is the same as b times a is more important than the procedural process of memorizing vocabulary.

Let me just point you to Justin Reich’s post on The MTT2K Prize he and I are co-sponsoring and co-judging. I only want to add a +1 and maybe a smiley face next to this sentence:

As far as I’m concerned, MTT2K has brought all kinds of good to the world.

I’d like to see some more of the kind of engagement we saw this last week, the kind where online criticism turns into improved outcomes for millions of students in the span of 24 hours. I’m excited to see what comes of this.

This strikes me as a really, really effective way to assess the pedagogical content knowledge of new teachers: critique the pedagogy of the Khan Academy video of your choice. You could write an essay and add timecodes for reference or you and a friend could sit in front of the screen MST3K-style and snark your way through Khan’s lecture like John Golden and David Coffey.

I’m really curious how the Church of Our Lady of Technology in Silicon Valley will react to this kind of critique. That church tends to write off most educators’ criticism of Khan Academy as some admixture of jealousy and entrenchment. They aren’t always wrong about that. But the criticism that “this is actually fairly poor lecturing that’ll leave students with shaky procedural understanding and even shakier conceptual understanding” is much harder to refute. It’s also a difficult criticism to illustrate for people who aren’t teachers. This is the best illustration of that critique I’ve seen.

BTW. The low-rent production values don’t do justice to the quality of their concept and critique, though. Thirty dollars on sound equipment would go a long way towards making this a series math supervisors around the US would make required viewing for their inservice teachers.

2012 Jun 20. Kent Haines has eagle eyes and points out that Khan Academy pulled their video within a couple hours of this post. Christopher Danielson asks the right question, I think. Are they pulling the video to correct the mathematical errors, the pedagogical errors, or both. It’s one thing to mistakenly refer to the transitive property when you mean the commutative property. It’s another to teach students that multiplying integers requires the memorization of a bunch of rules that look like magic but just memorize them because okay?

2011 Jun 21. I had high hopes for that comments thread but it wobbled off course pretty fast.

2011 Jun 22. A reader e-mailed asking what kind of audio setup I’d recommend. Here’s what I wrote back:

There are lots of configurations that’ll serve our needs here and probably several that are cheaper or less cumbersome than the one I use to record audio of myself in presentations and lectures. Lately, though, I record video using whatever I have on hand. Then for audio I use:

Then I sync the audio and video in post. Here’s a video explaining the setup.

2012 Jun 22. Khan Academy has re-uploaded the video and the difference is stark. The new version is oriented towards conceptual understanding whereas the last offered you the bare minimum necessary to pass a multiple-choice test or keep your teacher and parents off your back.

I don’t know much about history (” … and Nowak begat Townsley, father of Cornally … “) but here are a couple of observations from a few years of watching math edubloggers come and go.

There are a few crude but useful ways to categorize math edubloggers. Some stay. Others quit. Some blog regularly. Others blog sporadically. Some bloggers construct posts while they’re teaching. Others construct posts after they’ve taught. The first two are fairly obvious, I suppose. The last one is the most interesting to me. You’ll find bloggers who include photos, student work, and other classroom artifacts in their posts as a matter of routine. These bloggers were developing those blog posts — maybe consciously, maybe subconsciously — at the same time they were developing those lesson plans.

Speaking personally, I realized one day that without intending to I had developed a critical community around my blog, a group of people who were willing to save me from my own lousy classroom design choices. They got better at giving criticism and I got better at receiving it. I also got better at posting the kind of rich, multimedia artifacts of classroom practice — photos, videos, handouts, etc. — that facilitated that criticism. I started to plan lessons while wondering at the same time, “What about this is gonna be worth sharing?” Lesson planning and blogging became hopelessly and wonderfully tangled up.

There are generations of math education bloggers that stick together in fascinating ways. Perhaps it goes without saying that math edubloggers start by reading blogs, then commenting on blogs they read, then writing their own blogs. (Not unlike every other kind of blogger, I suppose.) It’s interesting for me to lurk around, though, and see where the new bloggers are commenting. Totally anecdotally, new bloggers seem to interact primarily with a) bloggers who started blogging in their same generation and b) bloggers who started blogging in the previous generation. Confusing? Andrew Stadel and Fawn Nguyen both started blogging about math education at about the same time. They both provoke and encourage each other on their blogs. I also see them interact with Christopher Danielson who started blogging a little over a year earlier. Meanwhile, they comment less often on this blog because, I dunno, I’m some kind of old timer and they’re ageist or something. Basically: new bloggers find community at their own level of experience and they find mentors one limb above them in the math education blogging family tree.

Those are my only observations that aren’t completely obvious. It’s a weird community that is always hungry for personality and wisdom, that occasionally collaborates and supports itself in spectacular ways that knock the wind out of me.

Your summer assignment: jump in.

2012 Jun 18. Matt Townsley has a Google survey which may help us construct a family tree. I added my details. Feel free to pitch in.